We prove that for d ≥ 3, the 1-skeleton of any (d− 1)-dimensional doubly Cohen-Macaulay (abbreviated 2-CM) complex is generically drigid. This implies that Barnette’s lower bound inequalities for boundary complexes of simplicial polytopes ([4],[3]) hold for every 2-CM complex of dimension ≥ 2 (see Kalai [8]). Moreover, the initial part (g0, g1, g2) of the g-vector of a 2-CM complex (of dimensio...

If a pure simplicial complex is partitionable, then its h-vector has combinatorial interpretation in terms of any partitioning the complex. Given non-partitionable Δ, we construct Γ⊇Δ same dimension such that both Γ and relative (Γ,Δ) are partitionable. This allows us to rewrite as difference two h-vectors partitionable complexes, giving an analogous By contrast, for given Δ it not always possi...